Abstract
Given a compact basic semi-algebraic set \({\mathbf{K}} \subset {\mathbb{R}}^n\) , a rational fraction \(f:{\mathbb{R}}^n\to{\mathbb{R}}\) , and a sequence of scalars y = (y α), we investigate when \(y_\alpha =\int_{\mathbf{K}} x^\alpha f\,d\mu\) holds for all \(\alpha\in{\mathbb{N}}^n\) , i.e., when y is the moment sequence of some measure fdμ, supported on K. This yields a set of (convex) linear matrix inequalities (LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral ∫ fdμ when the moments of μ are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures \(\nu,\mu\) with support contained in K, to have \(d\nu=f d\mu\) for some rational fraction f.
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Lasserre, J.B. The K-moment problem with densities. Math. Program. 116, 321–341 (2009). https://doi.org/10.1007/s10107-007-0118-4
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DOI: https://doi.org/10.1007/s10107-007-0118-4